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Approximations for Radiative Cooling and Heating in the Solar Chromosphere A&A 539, A39 (2012) Astronomy DOI: 10.1051/0004-6361/201118366 & c ESO 2012 Astrophysics Approximations for radiative cooling and heating in the solar chromosphere M. Carlsson1,2 and J. Leenaarts1,3 1 Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029 Blindern, 0315 Oslo, Norway e-mail: [email protected] 2 Center of Mathematics for Applications, University of Oslo, PO Box 1053 Blindern, 0316 Oslo, Norway 3 Sterrekundig Instituut, Utrecht University, Postbus 80 000, 3508 TA Utrecht, The Netherlands Received 31 October 2011 / Accepted 4 January 2012 ABSTRACT Context. The radiative energy balance in the solar chromosphere is dominated by strong spectral lines that are formed out of LTE. It is computationally prohibitive to solve the full equations of radiative transfer and statistical equilibrium in 3D time dependent MHD simulations. Aims. We look for simple recipes to compute the radiative energy balance in the dominant lines under solar chromospheric conditions. Methods. We use detailed calculations in time-dependent and 2D MHD snapshots to derive empirical formulae for the radiative cooling and heating. Results. The radiative cooling in neutral hydrogen lines and the Lyman continuum, the H and K and intrared triplet lines of singly ionized calcium and the h and k lines of singly ionized magnesium can be written as a product of an optically thin emission (dependent on temperature), an escape probability (dependent on column mass) and an ionization fraction (dependent on temperature). In the cool pockets of the chromosphere the same transitions contribute to the heating of the gas and similar formulae can be derived for these processes. We finally derive a simple recipe for the radiative heating of the chromosphere from incoming coronal radiation. We compare our recipes with the detailed results and comment on the accuracy and applicability of the recipes. Key words. radiative transfer – Sun: chromosphere 1. Introduction chromosphere. We also get heating in the chromosphere from coronal radiation impinging from above. We will here address all The chromospheric radiative energy balance is dominated by three processes: cooling from lines and continua, heating from a small number of strong lines from neutral hydrogen, singly the same transitions and heating through absorption of coronal ionized calcium and singly ionized magnesium (Vernazza et al. radiation. 1981) although a large number of iron lines may be equally We use detailed radiative transfer calculations from dynami- important in the lower atmosphere (Anderson & Athay 1989). cal snapshots to derive simple lookup tables that only use easily The strong lines are formed out of Local Thermodynamic obtainable quantities from a single snapshot from a simulation. Equilibrium (LTE) which means that numerical simulations of We test these recipes by applying them to several simulation runs the dynamic chromosphere need to solve simultaneously the and compare the radiative losses from detailed solutions with (magneto-)hydrodynamic equations, radiative transfer equations those from the recipes and find in general good agreement. and rate equations for all radiative transitions and energy lev- The structure of the paper is as follows: in Sect. 2 we els involved. This is a major computational effort that has been describe the hydrodynamical snapshots used for deriving the accomplished in one dimensional simulations of wave propa- lookup tables, in Sect. 3 we describe the atomic models used, gation in the quiet solar atmosphere (Klein et al. 1976, 1978; in Sect. 4 we derive the recipes for radiative cooling from strong Carlsson & Stein 1992, 1994, 1995, 1997, 2002; Rammacher & lines of hydrogen, calcium and magnesium, in Sect. 5 we de- Ulmschneider 2003; Fossum & Carlsson 2005, 2006), sunspots rive the recipes for heating of cool pockets of gas from the same (Bard & Carlsson 2010), solar flares (Abbett & Hawley 1999; lines, in Sect. 6 we compare the recipes with a detailed calcula- Allred et al. 2005; Kašparová et al. 2009; Cheng et al. 2010)and tion, in Sect. 7 we derive the recipes for the heating by incident stellar flares (Allred et al. 2006). coronal radiation and in Sect. 8 we end with our conclusions. In three-dimensional simulations the computational work re- quired to solve the full set of equations becomes prohibitive; there is a need for a simplified description that retains as much as 2. Atmospheric models possible of the basic physics but brings the computational work To calculate the semi-empirical recipes we use two sets of hy- to tractable levels. The aim of this paper is to provide such sim- drodynamical snapshots. The first is a simulation of wave propa- plified recipes for the chromospheric radiative energy balance. gation in the solar atmosphere using the RADYN code (Carlsson In addition to the cooling in strong lines and continua the & Stein 1992, 1994, 1995, 1997, 2002). RADYN solves the same transitions may provide heating of cool pockets in the one-dimensional equations of conservation of mass, momentum, Article published by EDP Sciences A39, page 1 of 10 A&A 539, A39 (2012) energy and charge together with the non-LTE radiative transfer and population rate equations, implicitly on an adaptive mesh. The grid positions of the adaptive mesh are determined by a grid equation that specifies where high resolution is needed (typically 4p 4p, where there are large gradients). The grid equation is solved si- 8498 8542 multaneously with the other equations. The simulation includes 8662 a 5-level plus continuum hydrogen atom, 5-levels of singly ion- ized calcium and one level of doubly ionized calcium and a 9-level helium atom including 5 levels of neutral helium, 3 of 3d 3d, 3933 singly ionized helium and one level of doubly ionized helium. 3968 All allowed radiative transitions between these levels are in- cluded. At the bottom boundary we introduce a velocity field that was determined from MDI observations of the Doppler shifts of the 676.78 nm Ni i line. Coronal irradiation of the chromosphere 4s was included using the prescription of Wahlstrom & Carlsson 2S 2P 2D 1S ff (1994) based on Tobiska (1991). The e ects of non-equilibrium ii ionization are handled self-consistently. The simulation is es- Fig. 1. Term diagram of the Ca model atom. The continuum is not shown. Energy levels (horizontal lines), bound-bound transitions (solid) sentially the one used in Judge et al. (2003) but with updated and bound-free transitions (dashed). atomic data for calcium. A number of ingredients that are poten- tially important for the chromospheric energy balance are miss- ing: line blanketing is not included (especially the iron lines may be important, Anderson & Athay 1989), cooling in the strong 3. Atomic models Mg ii h and k lines is not included and the description of partial redistribution is very crude for hydrogen and altogether missing 3.1. Hydrogen for calcium. Most important is probably the restriction of only one dimension. This prevents shocks from spreading out, forces We take the population densities directly from the RADYN sim- shock-merging of overtaking shocks and prevents the very low ulation and thus use the same atomic parameters. We use a five- temperatures found in 2D and 3D simulations from strong ex- level plus continuum model atom with energies from Bashkin pansion in more than one dimension. & Stoner (1975), oscillator strengths from Johnson (1972), pho- toionization cross-sections from Menzel & Pekeris (1935)and The other snapshot is from a 2D simulation of the solar atmo- collisional excitation and ionization rates from Vriens & Smeets sphere spanning from the upper convection zone to the corona. (1980) except for Lyman-α where we use Janev et al. (1987). This simulation was performed with the radiation-magneto- We use Voigt profiles with complete redistribution (CRD) for hydrodynamics code Bifrost (Gudiksen et al. 2011) and includes all lines. For the Lyman lines, profiles truncated at six Doppler ff the e ects of non-equilibrium ionization of hydrogen using the widths are used to mimic effects of partial redistribution (PRD) methods described in Leenaarts et al. (2007). A detailed de- (Milkey & Mihalas 1973). scription of the setup and results of this simulation is given in Leenaarts et al. (2011). Also here we have shortcomings in the description. Line blanketing is included but with a two- 3.2. Calcium level formulation for the scattering probability, hydrogen non- We use a five-level plus continuum model atom. It contains the equilibrium ionization is included but the excitation balance ii iii (needed to calculate the individual lines) is very crude. five lowest energy levels of Ca and the Ca continuum (see Fig. 1). The energies are from the NIST database with data com- Both simulations thus have their shortcomings. The purpose ing from Sugar & Corliss (1985), the oscillator strengths are of this paper is, however, to develop recipes that reproduce the from Theodosiou (1989) with lifetimes agreeing well with the results of detailed non-LTE calculations. It is thus not crucial experiments of Jin & Church (1993) and the broadening pa- that the simulations reproduce solar observations in detail as rameters are from the Vienna Atomic Line Database (VALD) long as they cover the parameter space expected in the solar (Piskunov et al. 1995; Kupka et al. 1999). Photoionization chromosphere. cross-sections are from the Opacity project. Collisional excita- For hydrogen it is important to include the coupling be- tion rates are from Meléndez et al. (2007) extrapolated beyond tween the non-equilibrium ionization and the non-LTE excita- the tabulated temperature range of 3000 K to 38 000 K using tion and we therefore use the RADYN simulation for the hydro- Burgess & Tully (1992). Collisional ionization rates from the gen recipes. For magnesium and calcium it is more important ground state are from Arnaud & Rothenflug (1985) and from to include PRD and have a large temperature range than to in- excited states we use the general formula provided by Burgess clude effects of non-equilibrium ionization and we choose the & Chidichimo (1983).
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